In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained, which turn out to be sharp. These rates coincide with rates obtained by numerical experiments, which are included in the paper.
Mots clés : variational inequalities, optimal control, control constraints, regularization, C-stationarity, path-following
@article{M2AN_2013__47_3_771_0,
author = {Schiela, Anton and Wachsmuth, Daniel},
title = {Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {771--787},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/m2an/2012049},
mrnumber = {3056408},
zbl = {1266.65112},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012049/}
}
TY - JOUR AU - Schiela, Anton AU - Wachsmuth, Daniel TI - Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 771 EP - 787 VL - 47 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012049/ DO - 10.1051/m2an/2012049 LA - en ID - M2AN_2013__47_3_771_0 ER -
%0 Journal Article %A Schiela, Anton %A Wachsmuth, Daniel %T Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 771-787 %V 47 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012049/ %R 10.1051/m2an/2012049 %G en %F M2AN_2013__47_3_771_0
Schiela, Anton; Wachsmuth, Daniel. Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 771-787. doi : 10.1051/m2an/2012049. http://www.numdam.org/articles/10.1051/m2an/2012049/
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